Non-wandering points and the depth for graph maps

Abstract

Let G be a graph and f: G → G be continuous. Denote by R(f) and Ω(f) the set of recurrent points and the set of non-wandering points of f respectively. Let Ω0(f) = G and Ωn(f) = \(\Omega (f|_{\Omega _{n - 1} (f)} )\) for all n ∈ ℕ. The minimal m ∈ ℕ ∪ {∞} such that Ωm(f) = Ωm+1(f) is called the depth of f. In this paper, we show that \(\Omega _2 (f) = \overline {R(f)} \) and the depth of f is at most 2. Furthermore, we obtain some properties of non-wandering points of f.